Discrete Time Instants Research Articles

Formula omitted] filtering with stochastic sampling

This paper is concerned with the problem of H ∞ filtering for continuous-time systems under sampled measurements with probabilistic sampling. It is assumed that the occurrence probabilities of the sampling intervals are given constants and satisfy a Bernoulli distribution. Through a transformation of the discrete time instants, the filtering error system is formulated as a continuous-time system with delays and stochastic parameters. Then, H ∞ filter is designed such that the filtering error system is exponentially stable in the mean square, and the L 2 -induced gain from the noise signal to the estimation error is guaranteed to be less than a prescribed level. Finally, an example is given to show the effectiveness of the theoretical results.

Networked Control of Distributed Energy Resources: Application to Solid Oxide Fuel Cells

This paper presents a model-based networked control approach for managing distributed energy resources (DERs) over communication networks. As a model system, we consider a solid oxide fuel cell (SOFC) plant that communicates with the central controller over a bandwidth-constrained communication network that is shared by several other DERs. The objective is to regulate the power output of the fuel cell while keeping the communication requirements with the controller to a minimum in order to reduce unnecessary network utilization and minimize the susceptibility of the SOFC plant to possible communication disruptions in the network. Initially, a feedback control law is designed to regulate the power output of the SOFC plant at a desired set-point by manipulating the inlet fuel flow rate. Network utilization is then reduced by minimizing the rate of transfer of information between the fuel cell sensors and the central controller without sacrificing the desired stability or performance properties. To this end, a dynamic model of the SOFC plant is embedded in the controller to approximate the dynamics of the plant when measurements are not transmitted by the sensors, and the state of the model is updated using the actual state that is provided by the SOFC plant sensors at discrete time instances. When full-state measurements are not available, an appropriate state observer is included in the control structure to generate state estimates from the measured outputs, which are then used to update the model states. An explicit characterization of the maximum allowable transfer time between the sensor suite of the SOFC plant and the controller (i.e., the minimum allowable communication rate) is obtained under both state and output feedback control in terms of plant-model mismatch and the choice of control law. The characterization accounts for both stability and performance considerations. Finally, numerical simulations that demonstrate the implementation of the networked control architecture and its disturbance handling capabilities are presented.

Problem of reconstructing input signals under communication constraints

For a linear controllable system, the problem of reconstruction of all input signals, which are compatible with the measured signal, is considered. It is assumed that information can be transmitted in a processing center only via digital communication channel at discrete time instants, and the word length is limited. In this connection, there are encoding and decoding devices in the communication channel. For simplicity, the communication channel is assumed to be noiseless and delay-free. Defining relationships are obtained for the set of compatible input signals, as well as relationships between reconstruction precision, length of transmitted word and transmission frequency.

H ∞ fuzzy filtering design for non-linear sampled-data systems

The problem of H∞ filtering design is studied for non-linear sampled-data systems using the Takagi–Sugeno (T–S) fuzzy model approach. The sampled-data filtering is to estimate the states of a continuous-time system using only sampled measurements at discrete instants of time. Traditionally, the sufficient conditions for the existence of such an H∞ filter are characterised in terms of the solution of a differential Hamilton–Jacobi inequality with jumps, which is equivalent to solving the partial differential inequality with jumps. In general, there is no analytical solution for this non-linear partial differential inequality with jumps. First, in this study, the T–S fuzzy model is proposed to represent a class of non-linear sampled-data systems. Next, by using the T–S fuzzy model, the H∞ fuzzy filtering design problem for non-linear sampled-data system is characterised in terms of a linear matrix inequality (LMI) problem. Hence, the H∞ fuzzy filter of non-linear sampled-data systems can be given via solving LMIs instead of solving a differential Hamilton–Jacobi inequality with jumps. To illustrate the results, a numerical example is included.

On the singularity of least squares estimator for mean-reverting α-stable motions

We study the problem of parameter estimation for mean-reverting a-stable motion, dXt = (a0 - ϑ0Xt)dt + dZt observed at discrete time instants. A least squares estimator is obtained and its asymptotics is discussed in the singular case (a0, ϑ0) = (0, 0). If a0 = 0, then the mean-reverting a-stable motion becomes Ornstein-Uhlenbeck process and is studied in [7] in the ergodic case ϑ0 > 0. For the Ornstein-Uhlenbeck process, asymptotics of the least squares estimators for the singular case (ϑ0 = 0) and for ergodic case (ϑ0 > 0) are completely different.

Exact Discretization of a Matrix Differential Riccati Equation With Constant Coefficients

An exact method is presented for discretizing a constant-coefficient, non-square, matrix differential Riccati equation, whose solution is assumed to exist. The resulting discrete-time equation gives the values that have no error at discrete-time instants for any discrete-time interval. The method is based on a matrix fractional transformation, which is more general than existing ones, for linearizing the differential Riccati equation. A numerical example is presented to compare the proposed method with that based on gage invariance and bilinearization, which has better performances than the conventional forward-difference method.

Stochastic stability of Markovian switching genetic regulatory networks

In this Letter, taking into account the structure variations at discrete time instances during the process of gene regulation, a hybrid genetic regulatory networks model based on Markov chain is proposed. Its robust stochastic stability in the case of uncertain switching probabilities and intrinsic noises is then addressed from the stochastic system point of view. It is shown that the sufficient condition for the robust stochastic stability of the genetic networks can be formulated as feasibility of a linear matrix inequality, which can be easily facilitated by Matlab LMI toolbox. Finally, a numerical example with simulations is presented to illustrate the effectiveness of the developed results.

Multi-Objective Evolutionary Optimization of Biological Pest Control with Impulsive Dynamics in Soybean Crops

The biological pest control in agriculture, an environment-friendly practice, maintains the density of pests below an economic injury level by releasing a suitable quantity of their natural enemies. This work proposes a multi-objective numerical solution to biological pest control for soybean crops, considering both the cost of application of the control action and the cost of economic damages. The system model is nonlinear with impulsive control dynamics, in order to cope more effectively with the actual control action to be applied, which should be performed in a finite number of discrete time instants. The dynamic optimization problem is solved using the NSGA-II, a fast and trustworthy multi-objective genetic algorithm. The results suggest a dual pest control policy, in which the relative price of control action versus the associated additional harvest yield determines the usage of either a low control action strategy or a higher one.

Distributed discrete-time coordinated tracking with a time-varying reference state and limited communication

This paper studies a distributed discrete-time coordinated tracking problem where a team of vehicles communicating with their local neighbors at discrete-time instants tracks a time-varying reference state available to only a subset of the team members. We propose a PD-like discrete-time consensus algorithm to address the problem under a fixed communication graph. We then study the condition on the communication graph, the sampling period, and the control gain to ensure stability and give the quantitative bound of the tracking errors. It is shown that the ultimate bound of the tracking errors is proportional to the sampling period. The benefit of the proposed PD-like discrete-time consensus algorithm is also demonstrated through comparison with an existing P-like discrete-time consensus algorithm. Simulation results are presented as a proof of concept.

Least squares estimator for Ornstein–Uhlenbeck processes driven by [formula omitted]-stable motions

We study the problem of parameter estimation for generalized Ornstein–Uhlenbeck processes driven by α -stable noises, observed at discrete time instants. Least squares method is used to obtain an asymptotically consistent estimator. The strong consistency and the rate of convergence of the estimator have been studied. The estimator has a higher order of convergence in the general stable, non-Gaussian case than in the classical Gaussian case.

Integrating Control and Scheduling of Distributed Energy Resources Over Networks

This paper presents an integrated approach for the control and scheduling of Distributed Energy Resources (DERs) that are managed by a central supervisor over a resource-constrained communication network. The objective is to enhance the performance and disturbance-handling capabilities of the DERs while keeping the communication requirements with the supervisor to a minimum in order to reduce the susceptibility of the DERs to communication outages. To this end, the rate of data transfer from the DERs to the supervisor is initially minimized by embedding in the supervisor a set of models that are used to generate the necessary control action when measurements are not transmitted over the network, and then updating the models' states at discrete time instances. Only a subset of the DERs are allowed to transmit their data at any given time to provide updates to their target models according to a certain scheduling strategy. By formulating the networked closed-loop system as a hybrid system, an explicit characterization of the interdependence between the performance of the DERs, the communication rate, the transmission schedule and times, and the plant-models' mismatch is obtained. It is shown that by judicious selection of the transmission schedule and models, it is possible to optimize the performance of the DERs while simultaneously reducing network utilization beyond what is possible with concurrent transmission configurations. The results are demonstrated through an application to a collection of solid oxide fuel cells in a distributed power network.

Quasi-decentralized Scheduled Output Feedback Control of Process Systems Using Wireless Sensor Networks

This paper presents a quasi-decentralized output feedback control structure for multi-unit plants with limited state measurements and distributed control systems that exchange information over a resource-constrained wireless sensor network (WSN). The networked control structure brings together model-based feedback control, state estimation and sensor scheduling to enforce closed-loop stability while simultaneously minimizing the rate of communication over the WSN. Initially, an observer-based output feedback controller is designed for each unit. To conserve the resources of the wireless devices, communication between the local control systems is suspended periodically for extended time periods during which each control system relies on models of the plant units to generate the necessary control action. Communication is then re-established at discrete time instances according to a certain schedule that determines the order and times at which the wireless sensor suites transmit the state estimates needed to update the states of the models embedded in the target units. By analyzing the combined discrete-continuous behavior of the scheduled closed-loop plant, we explicitly characterize the stability of the networked closed-loop system in terms of the communication rate, the sensor transmission schedule, the accuracy of the models, as well as the controller and observer design parameters. The results are illustrated using a chemical plant example where it is shown that by judicious management of the interplays between the control, communication and scheduling design parameters, it is possible to stabilize the plant while simultaneously enhancing the savings in WSN resources beyond what is possible with concurrent transmission configurations.

Application of Girsanov theorem to particle filtering of discretely observed continuous-time non-linear systems

This article considers the application of particle filtering to continuous-discrete optimal filtering problems, where the system model is a stochastic differential equation, and noisy measurements of the system are obtained at discrete instances of time. It is shown how the Girsanov theorem can be used for evaluating the likelihood ratios needed in importance sampling. It is also shown how the methodology can be applied to a class of models, where the driving noise process is lower in the dimensionality than the state and thus the laws of the state and the noise are not absolutely continuous. Rao-Blackwellization of conditionally Gaussian models and unknown static parameter models is also considered.

Dynamic 3D axisymmetric problems in continuously non-homogeneous piezoelectric solids

The meshless local Petrov-Galerkin (MLPG) method is used to analyze transient dynamic problems in 3D axisymmetric piezoelectric solids with continuously inhomogeneous material properties. Both mechanical and thermal loads are considered here. A 3D axisymmetric body is created by rotation of a cross section around an axis of symmetry. Axial symmetry of geometry and boundary conditions reduces the original 3D boundary value problem into a 2D problem. The cross section is covered by small circular sub-domains surrounding nodes randomly spread over the analyzed domain. A unit step function is chosen as test function, in order to derive local integral equations on the boundaries of the chosen sub-domains, called local boundary integral equations (LBIE). These integral formulations are either based on the Laplace transform technique or the time-difference approach. The local integral equations are non-singular and take a very simple form, despite of inhomogeneous and anisotropic material behaviour across the analyzed structure. Spatial variation of all physical fields (or of their Laplace transforms) at discrete time instants are approximated on the local boundary and in the interior of the sub-domain by means of the moving least-squares (MLS) method. The Stehfest algorithm is applied for the numerical Laplace inversion, in order to retrieve the time-dependent solutions.

Quasi-decentralized model-based networked control of process systems

This paper develops a quasi-decentralized control framework for plants with distributed, interconnected units that exchange information over a shared communication network. In this architecture, each unit in the plant has a local control system that communicates with the plant supervisor – and with other local control systems – through a shared communication medium. The objective is to design an integrated control and communication strategy that ensures the desired closed-loop stability and performance for the plant while minimizing network utilization and communication costs. The idea is to reduce the exchange of information between the local control systems as much as possible without sacrificing stability of the individual units and the overall plant. To this end, dynamic models of the interconnected units are embedded in the local control system of each unit to provide it with an estimate of the evolution of its neighbors when measurements are not transmitted through the network. The use of a model to recreate the interactions of a given unit with one of its neighbors allows the sensor suite of the neighboring unit to send its data in a discrete fashion since the model can provide an approximation of the unit’s dynamics. The state of each model is then updated using the actual state of the corresponding unit provided by its sensors at discrete time instances to compensate for model uncertainty. By formulating the networked closed-loop plant as a hybrid system, an explicit characterization of the maximum allowable update period (i.e., minimum cross communication frequency) between each control system and the sensors of its neighboring units is obtained in terms of the degree of mismatch between the dynamics of the units and the models used to describe them. The developed control strategy is illustrated using a network of interconnected chemical reactors with recycle.

A Lavrent'ev-type approach to the on-line computation of Caputo fractional derivatives**This paper fits the research programs of GNAMPA-INDAM.

The computation of Caputo fractional derivatives is an ill-posed problem (in fact a special deconvolution problem) which can be approached using suitable regularization methods. Recent applications of deconvolution problems in control theory require recursive regularization methods. For this reason we present here a recursive method for the computation of an approximant vϵ(t) of the Caputo fractional derivative (Dα*f)(t), which is an extension of the usual Lavrent'ev method (ϵ is the regularization parameter). In applications to control theory the signal f whose derivative has to be computed is measured at discrete time instants kτ. In this case the algorithm we propose updates the value of vϵ at the times kτ when the new measure of the signal f(t) is received. The algorithm can be applied to fractional (and integer) derivatives of any order.

An Observer that Converges in Finite Time Due to Measurement-based State Updates

This paper presents a new observer that estimates the exact state of a linear continuous-time system in predetermined finite time. The finite convergence time of the proposed observer is achieved by updating the observer state based on the difference between the measured output and the estimated output at discrete time instants. Simulation results are presented to illustrate the convergence behavior and the applicability of the proposed observer.

Sampled-Data Filtering Framework for Cardiac Motion Recovery: Optimal Estimation of Continuous Dynamics From Discrete Measurements

Quantitative and noninvasive estimation of cardiac kinematics has significant physiological and clinical implications. In this paper, a sampled-data filtering framework is presented for the recovery of cardiac motion and deformation functions from periodic medical image sequences. Cardiac dynamics is a continuously evolving physical/physiological process, whereas the imaging data can provide only sampled measurements at discrete time instants. Given such a hybrid paradigm, stochastic multiframe filtering frameworks are constructed to couple the continuous dynamics with the discrete measurements, and to coordinately deal with the parameter uncertainty of the biomechanical constraining model and the noisy nature of the imaging data. The state estimates are predicted according to the continuous-time biomechanically constructed state equation between observation time points, and then updated with the new imaging-derived measurements at discrete time instants, yielding physically more meaningful and more accurate estimation results. Both continuous-discrete Kalman filter and sampled-data Hinfinity filter are applied for motion recovery. While Kalman filter is the optimal estimator under Gaussian noises, the Hinfinity scheme can give robust estimation results when the types and levels of model uncertainties and data disturbances are not available a priori. The strategies are validated through synthetic data experiments to illustrate their advantages and on canine MR phase contrast images and human MR tagging data to show their clinical potential.

Optimum experimental designs for dynamic systems in the presence of correlated errors

The problem of determining an optimal measurement time schedule for identification of unknown parameters in multiresponse systems when correlations between observations occur is considered. The measurement process is performed by collecting data at discrete time instants from several outputs. An observation plan is proposed based on a scalar measure of the Fisher information matrix as the design criterion quantifying the accuracy of parameter estimators. A numerical procedure is proposed to determine approximations of optimum designs in the case of correlated measurement errors. The approach is illustrated with an example of the multi-output system of equations describing a chemical kinetic reaction.

A statistical approach to simultaneous mapping and localization for mobile robots

Mobile robots require basic information to navigate through an environment: they need to know where they are (localization) and they need to know where they are going. For the latter, robots need a map of the environment. Using sensors of a variety of forms, robots gather information as they move through an environment in order to build a map. In this paper we present a novel sampling algorithm to solving the simultaneous mapping and localization (SLAM) problem in indoor environments. We approach the problem from a Bayesian statistics perspective. The data correspond to a set of range finder and odometer measurements, obtained at discrete time instants. We focus on the estimation of the posterior distribution over the space of possible maps given the data. By exploiting different factorizations of this distribution, we derive three sampling algorithms based on importance sampling. We illustrate the results of our approach by testing the algorithms with two real data sets obtained through robot navigation inside office buildings at Carnegie Mellon University and the Pontificia Universidad Catolica de Chile.